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Apr 1, 2017 – Dexter The Complete Series On Blu-ray (2013-17).Q:

proving $\sum \frac{1}{n^3}$ diverges

I have to show this sum
$$\sum_{n=1}^{\infty}\frac{1}{n^3}$$
diverges.
I know it has something to do with Cauchy condensation test and I know I have to use the harmonic series here.
Thanks for any help.

A:

For a proof that the harmonic series diverges, use the fact that
$$\sum_{n=1}^{\infty}\frac{1}{n^2}=\sum_{n=1}^{\infty}2\left(\frac{1}{2n}+\frac{1}{2n+1}\right)=2+2\sum_{n=1}^{\infty}\frac{1}{n}=\infty.$$

A:

Here is another approach. Let $f(x)=\frac{1}{x^3}$ for $x>0$. Then $f$ is continuous on $[0,\infty)$ and $f'(x) = \frac{3}{x^4} 0$. So by the mean value theorem, for any $N>0$, we have $\frac{f(N)}{N}= f'(\xi_N)N = N f'(\xi_N) = N \frac{3}{\xi_N^4}<0$ so that $f(N)<0$ and $f(x)$ is decreasing. It follows that $\lim_{x \to \infty} f(x)=-\infty$. Thus, $f$ is not integrable over $[0,\infty)$, and so $\sum_{n=1}^\infty\frac{1}{n^3}$ diverges.

A:

Hint:
Use the fact
$$\sum_{n=1}^\infty\frac{1}{n^3}=\sum_{n=1}^\infty\frac{1}{n^2}-\sum_{n=1}^\infty\frac{

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